A carrot is a root vegetable. And while fruits and veggies are all generally better than, say, Reese's Peanut Butter Cups, they're not all created equal. Mushrooms, for example, don't have much worth writing home about. Like Phate said, sugar, carbs and nutrients vary in each. So you can't just eat potatoes and strawberries and call it good.

I recommend checking out the website whfoods.com. There is a lot of information about what makes certain fruits and vegetables (and nuts and herbs and meat and etc.) healthy.

The only fatty fruits/vegetables I can think of are olives, avocados, and coconuts. And of course, olive oil and avocados are both great sources of healthy monounsaturated fat. Coconut is of course full of saturated fat and should be consumed in moderation, but its saturated fatty acids are supposedly shorter chain than, say, butter or beef. So I wouldn't even avoid coconut completely. I just would not consume too much of it.

Some fruits/vegetables have little in the way of nutrition, such as iceberg lettuce. They might help you feel full on fewer Calories, and they might provide some of the fiber you need, but there are certainly better choices out there, such as romaine lettuce, spinach, or swiss chard. Some fruits and vegetables also have lots of carbs. But pretty much all of them are reasonably healthy so long as you don't deep-fry them.

Some foods, such as onions and tomatoes, supposedly are healthier when they are somewhat cooked than when they are raw. Many though, such as green vegetables, have nutrients that are damaged by heat. With such vegetables, it's better to gently steam them, through them into a stir-fry near the end, or not cook them at all.